Counting Liberties

Part 1

Masayoshi Fukuda, 6p

The following article is intended for relatively experienced
players, who frequently have difficulty in determining how a fight will
turn out because they dont know some of the formulae that more skilful
players use.

Let us first consider Dia 1a. The two chains of 3 stones cannot both
survive. In this simple example it is fairly easily seen that whoever
attacks first will win. The least experienced player has learned that in
simple situations like this he need only count the number of vacant
points to which each group connects, and that the first play wins if the
number is equal.

He should also know that, if his group has one more liberty than his
opponents, then he can play elsewhere - there will be enough time if his
opponent attacks first. Thus in Dia 1b, the white three stones are
lost; Black can play elsewhere and need not take action here until White
plays at one of the 4 liberties of the 7 stone black group. All Black
needs do is count the liberties - he has 4, White has 3.

Although in simple situations the fighting power of a group is
determined by the number of vacant points to which it connects, this is
no longer valid in more complicated positions. From now on the tern
"liberties" will be used to denote the fighting power of a group. An
example will serve to clarify this point.

Consider Dia 1c. The cut-off black chain cannot live except by
capturing the three white stones to the right - the capture of the
three other stones does not lead to two eyes, since White will play back
inside. Similarly, the three white stones to the right can only survive
by killing the cut-off black chain. These whites have three liberties;
how many has the black chain? If we play the sequence out, assuming that
White starts, it will go as in Dias 1d & 1e.

Black 2 captures 3 white stones. White 7 captures 9 black stones.
- Since Black came out just one move behind, we conclude that he
too must have had three liberties. The reader can check this by playing
the situation through giving Black the first move - in which case he
wins.

In Dias 2a thru 2d we see four positions in which white chains
enclose black stones, and are themselves enclosed by black groups. In
Dia 2a, which is similar to the situation we have just been considering,
three stones are enclosed; in B, four; in C, five; in D, six. In each
case white cannot make two eyes against opposition after capturing the
inside black stones. Such formations are painfully familiar to all
beginners; the reader should be able to recognise them at once. How many
liberties does the inside white group have in each case?

For the positions in Dia 1a thru 1c this inquiry is academic, since
for simplicity these white stones have no fighting chance, but academic
inquiries can be instructive.

Let us play through the sequence for Dia 2c (in Dias 2e thru 2j),
with Black playing first and indicating the White opportunities to
counter-attack, as distinguished from forced plays inside the group.

We count seven missed moves before the death of the white stones;
adding one we conclude that white had 8 liberties. If we go through a
similar sequence for each position, we arrive at the table below. A few
minutes spent in memorislng this table will save many anxious minutes of
mental gymnastics during games.

Captives

3

4

5

6

Liberties

3

5

8

12

It should be noted that in each case the trapped white stones have
one inside and one outside vacant point to which they connect, and that
each sequence starts with a play by the attacker filling the outside
liberty followed by the defender capturing the attacker's stones on the
inside.

Let us now consider some of the applications of the table, and then
discuss the situations where, appearances to the contrary, it does not
apply.

Dias 3a thru 3d show four positions where knowing the above table
permits us to say after a very brief examination that first play wins,
since in each case the fighting groups have an equal number of
liberties. Dia 3a we have already analysed (in Dia 1c to 1f). Three captives, giving
three liberties for Black; three obvious liberties for White.

In Dia 3b the 4:5 rule applies. White starts at 1, Black replies 2.
These two plays are like the first two plays in preceding examples. From
our table, then, we know that the Blacks have five liberties; the Whites
also have five. Whoever plays first wins.

In Dia 3c, the situation is similar, illustrating the 5:8 rule
- eight liberties for the Whites, and eight for the Blacks. In Dia
3d the 6:12 rule holds and the first player wins twelve liberties for
the Whites, twelve for the Blacks. How many players could reach this
conclusion about Dia 3d without knowing the 6:12 formula?

To be concluded.

This article is from the
British Go Journal
Issue 10
which is one of a series of back issues now available on the web.